How might mathematics be different today if we were using Frege's set theory instead of ZFC?

Question

Soft question: How might mathematics be different today if we were using Frege's set theory instead of ZFC? Assume no inconsistencies like Russell's Paradox had ever been discovered in Frege's set theory.

Answer 1

Not much, if at all. First of all, most of "ordinary mathematics" does not depend on which formalization is used for set theory, ZFC, NBG or Russell's Principia. For that matter, Russell's paradox does not arise if Frege's Basic Law V is replaced by the weaker Hume's principle, which Frege knew and modern neo-Fregeans promote. So one can use Frege's second order logic for foundations too, "set theory in sheep's closing", as Quine called it. First order theories, like ZFC, won over higher order intensional theories for reasons of technical convenience, and they likely would have done so even if Russell's paradox was never discovered, and Russell's ramified type theory was not as onerous.

Even within mathematical logic Frege's influence was not as great as one might expect. He himself was not well versed in prior developments that influenced 19th century logic, and although he developed the predicate calculus with quantifiers slightly earlier than Peirce and Mitchell, who were so versed, it was their version, systematized in Schröder's Algebra of Logic, that was picked up by Russell and became common today. Here is Dipert's summary from Peirce, Frege, the Logic of Relations, and Church's Theorem (History and Philosophy of Logic, 5 (1984), 49-66):

However, Frege's battle was not lost. His victory was merely postponed a generation. Unfortunately, the legitimate contributions of Peirce and Schröder, especially to the logic of the relations, did get lost in the ensuing fray. Neither Peirce nor Schröder had the services of such an excellent propagandist as Russell. The Peirce-Schröder calculus was portrayed as purely algebraic, without the variable-binding operators Peirce regarded as essential and to which Schröder usually resorted; its weaknesses were rhetorically exploited with the bon mot "too complicated"; its subtlest achievements were ignored (e.g., clever theorems proven and Peirce's insights into the differences between the monadic and polyadic predicate logic); and, in a final injustice, the development of the theory of relations in Principia Mathematica owes most, especially in notation, to Schröder via the influence of Peano rather than to Frege, but it was presented without substantial acknowledgement. Alfred Tarski is one of the few logicians or historians writing in the 20th century who seems to realize the proportions of this injustice.

In fairness, Russell and subsequent logicians felt more affinity to Frege for a reason. His view of mathematical foundations and rigor was more in line with the one that eventiually emerged through the efforts of Peano, Hilbert, Zermelo, Russell, and many others. But again, this view is not specifically tied to Frege's choice of formalism. Here is Dipert again:

Frege's theory, even as presented in his earlier Begriffsschrift, is superior to later work of Peirce and Schröder in a number of respects. Frege's notion of a formal system, and his concern for foundations, are exemplary. By comparison, Peirce's and Schröder's proofs appear like much mathematics appears to logicians: sloppy, unrigorous, but in some sense 'correct'. In spite of his superior formal system, however, Frege seems not to have appreciated fully how much his account of quantified relational statements enriched logic... The weight of advancing Frege's case for his superior handling of quantified relational statements thus fell on Peano and Russell.